Child Development, July/August 2014, Volume 85, Number 4, Pages 1461–1476
Sources of Individual Differences in Children’s Understanding of Fractions
Rose K. Vukovic
Lynn S. Fuchs
New York University
Vanderbilt University
David C. Geary
Nancy C. Jordan
University of Missouri
University of Delaware
Russell Gersten
Robert S. Siegler
Instructional Research Group
Carnegie Mellon University & Beijing
Normal University
Longitudinal associations of domain-general and numerical competencies with individual differences in children’s understanding of fractions were investigated. Children (n = 163) were assessed at 6 years of age on
domain-general (nonverbal reasoning, language, attentive behavior, executive control, visual-spatial memory)
and numerical (number knowledge) competencies; at 7 years on whole-number arithmetic computations and
number line estimation; and at 10 years on fraction concepts. Mediation analyses controlling for general mathematics ability and general academic ability revealed that numerical and mathematical competencies were
direct predictors of fraction concepts, whereas domain-general competencies supported the acquisition of fraction concepts via whole-number arithmetic computations or number line estimation. Results indicate multiple
pathways to fraction competence.
Competence with fractions is an integral part of
overall mathematical proficiency, laying the foundation for learning the advanced mathematics that is
critical for obtaining full economic opportunity and
meaningful participation in society (e.g., Moses &
Cobb, 2001; National Mathematics Advisory Panel,
2008). Yet fractions represent a primary source of
difficulty for many students (Hecht, Close, & Santisi, 2003; National Mathematics Advisory Panel,
2008; Siegler et al., 2012; Tatsuoka, Corter, & Tatsuoka, 2004). For example, only 40% of fourth
graders in the United States can compare unit fractions—such as explaining whether 1/4 is smaller or
larger than 1/5 (National Center for Education
Statistics, 2007), and only half can identify pictorial
This research was supported in part by Award R01HD053714
from the Eunice Kennedy Shriver National Institute of Child
Health & Human Development (NICHD) to Vanderbilt University; by Grant R37 HD045914 from NICHD to the University of
Missouri; by Grant R324C100004 from the Institute of Education
Sciences in the U.S. Department of Education to the University
of Delaware with subcontracts to Vanderbilt University and to
Carnegie Mellon University; and by Core Grant HD15052 from
NICHD. The content is solely the responsibility of the authors
and does not necessarily represent the official views of the
NICHD, the National Institutes of Health, or the U.S. Department of Education.
Correspondence concerning this article should be addressed to
Rose K. Vukovic, 239 Greene St, 5th Floor, New York, NY 10003.
Electronic mail may be sent to [email protected].
representations of equivalent fractions (National
Center for Education Statistics, 2009). These difficulties remain well into high school (e.g., Calhoon,
Emerson, Flores, & Houchins, 2007). Furthermore,
difficulty with fractions is not only a U.S. phenomenon; several countries around the world report
fractions as challenging for students to master (e.g.,
Chan, Leu, & Chen, 2007; Tatsuoka et al., 2004).
Thus, a pressing need exists to improve understanding of the developmental precursors of
fraction knowledge, with the goal of providing theoretical insight into and practical guidance about
the identification and treatment of fraction difficulties.
This longitudinal study focused on identifying
early predictors of fourth-grade children’s understanding of core fraction concepts (e.g., fractions
represent the division of whole units into parts;
fractions can be used to describe proportions and
magnitudes; an infinite number of fractions exist
between any two values; Siegler & Pyke, 2013). We
focused specifically on fraction concepts because
conceptual understanding tends to have larger
© 2014 The Authors
Child Development © 2014 Society for Research in Child Development, Inc.
All rights reserved. 0009-3920/2014/8504-0012
DOI: 10.1111/cdev.12218
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Vukovic et al.
effects on the development of procedural proficiency and has more generalized effects on new
learning than does procedural knowledge (e.g.,
Hecht et al., 2003; Rittle-Johnson, Siegler, & Alibi,
2001).
around the same time that fractions become salient
in the school curriculum—or at other developmental stages, such as the transition to algebra, which
involves operating on functions rather than numbers, whole or rational.
Development of Fraction Competence
Theoretical Accounts of Fraction Learning
Children begin schooling with an informal
understanding of fraction concepts that does not
depend on knowledge of formal mathematical symbols. For instance, Mix, Levine, and Huttenlocher
(1999) found that 4-year-olds can reason with informal fraction problems involving solutions less than
or equal to one whole, whereas the ability to reason
with mixed-number fractions (i.e., > 1) emerges
around 6 years old. Mix et al. demonstrated that
children have at least an implicit understanding of
some fraction-related concepts long before fractions
are formally introduced in school. This intuitive
knowledge, however, does not easily translate into
symbolic knowledge of fractions. For example,
whereas young children have an informal understanding of the part–whole notion of fractions, not
until seventh grade do they consistently demonstrate an understanding of part–whole relations
with symbolic fraction notation (Stafylidou &
Vosniadou, 2004).
It has been suggested that learning formal fractions suffers in part because of what Ni and Zhou
(2005) call the whole-number bias. Many children
apply their understanding of whole-number properties when solving fraction problems (Stafylidou &
Vosniadou, 2004). This is problematic because, for
example, whole numbers follow a stable order
sequence (e.g., 5 always follows 4 in the counting
sequence), and each number in the sequence has a
greater magnitude than the previous number (e.g.,
5 is always larger than 4). Overgeneralization of
such properties contributes to difficulty in understanding, for example, that 1/4 is larger than 1/5.
At this time, the cognitive competencies that support the conceptual leap from whole numbers to
fractions are not well understood. Such understanding is critical not only given the importance of fraction competence for more advanced mathematics
and for competing in the U.S. workforce (National
Mathematics Advisory Panel, 2008; Siegler et al.,
2012) but also for generating hypotheses about the
processes by which such a conceptual leap is
achieved in fractions as well as other domains of
learning. This includes, for example, moving from
comprehension of stories to informational text—
another stumbling block for learners that occurs
Previous studies have shown that mathematical
skills are predicted by a combination of (a) domaingeneral competencies—the global processes that
contribute to cognitive development and learning
across several domains (e.g., attentive behavior,
executive control)—and (b) specific numerical competencies, such as the ability to represent and
manipulate small exact quantities (Fuchs, Geary,
Compton, Fuchs, Hamlett, Seethaler, et al., 2010;
Geary, 2011; LeFevre et al., 2010). Few studies,
however, have assessed the early competencies that
support the later development of fraction concepts,
even though this knowledge may have implications
for preventative work with young students to avoid
subsequent difficulty.
Two previous studies are particularly relevant in
specifying a theoretical model of fraction development. Hecht et al. (2003) proposed that domaingeneral competencies predict fraction knowledge
directly and indirectly through intervening mathematical skills hypothesized as necessary to become
proficient with fractions, such as fraction concepts
and whole-number computations. According to this
model, domain-general competencies influence
learning across all academic domains and thus support the acquisition of both fraction skill and intervening mathematical competencies. In a test of their
model with fifth graders, Hecht et al. found that
domain-general competencies were better predictors
of the intervening mathematical skills than of fraction skills. Specifically, attentive behavior (children’s ability to pay attention during instruction)
and executive control (the general purpose control
mechanism that regulates the maintenance and processing of cognitive subprocesses involved in immediate awareness; Miyake, Friedman, Emerson,
Witzki, & Howerter, 2000) were related to fraction
skills primarily through their association with the
intervening mathematical precursor skills. Furthermore, there was a differential relation between fraction skills and the intervening mathematical
competencies: Whereas fraction concepts were
related to all fraction skills (estimation, computations, and word problems), fluency with wholenumber combinations was related only to fraction
computations.
Understanding of Fractions
These findings indicate that domain-general and
mathematical competencies do not share uniform
relations with fraction learning. Moreover, Hecht
et al. (2003) tested the effects of only attentive
behavior and executive control, even though other
domain-general competencies, such as language
and nonverbal reasoning, have been implicated in
more recent research (e.g., Jordan et al., 2013; Seethaler, Fuchs, Star, & Bryant, 2011). Furthermore,
Hecht et al. considered mathematical competencies
only as intervening variables in the developmental
sequence of fraction learning. More recent studies
have shown that early numerical competencies,
including the ability to fluently process, represent,
and manipulate small exact quantities, are also
foundational for later mathematical development
independent of domain-general competencies (e.g.,
Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler,
et al., 2010; Geary, 2011; Geary, Bailey, & Hoard,
2009).
More recently, LeFevre et al. (2010) proposed an
alternative model of general mathematical development. The model proposes that three “pathways”
are involved in processing numerical information
and that these pathways play distinct roles in mathematical development. The linguistic pathway is a
domain-general system hypothesized to support the
development of symbolic representational systems,
including the symbolic number system. Thus, the
linguistic pathway should predict performance on
measures that require knowledge of the formal
number system, such as whole-number computations and fraction concepts. The quantitative pathway is a cognitive system involved in processing
numerosity independent of the symbolic number
system, such as through subitizing, or apprehending small collections of objects without counting.
The quantitative pathway is thought to support
the development of mathematical skills that require
children to represent and manipulate quantities,
including estimating quantities and understanding proportional magnitudes. Finally, the spatialattention pathway represents a complex set of
domain-general cognitive competencies—attentional
capacity, executive control, nonverbal reasoning,
visual-spatial memory—that are involved in processing numerical information separate from linguistic
and quantitative skills. As such, the spatial-attention
pathway is hypothesized to influence mathematical
outcomes generally versus having a distinct association with specific tasks.
In a longitudinal study with children from 4.5 to
7.5 years of age, LeFevre et al. (2010) found that
the three pathways differentially predicted various
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mathematical outcomes (i.e., whole-number computations, numeration, geometry, measurement, and
number line estimation). The linguistic pathway—a
composite including vocabulary, phonological skills,
and number identification—was uniquely related to
all mathematical outcomes, presumably because
these outcomes required knowledge of the symbolic number system. The quantitative pathway—a
composite including subitizing and a nonverbal
calculation task—was uniquely related to numeration, whole-number computations, and number line
estimation, ostensibly because these tasks require
numerical magnitude processing specifically. Finally,
the spatial-attention pathway, representing several
domain-general competencies, explained unique
variance across tasks tapping both the symbolic
number system and those tapping magnitude processing.
In thinking about the cognitive competencies that
may support the conceptual leap from whole-number to fraction understanding and how to design
interventions to support such a leap, the LeFevre
et al. (2010) model requires further specification for
two reasons. First, the composites used to operationalize the pathways do not permit a sufficiently
fine-grained analysis of specific processes. For
instance, a relation between the linguistic composite
and mathematical outcomes could reflect the influence of language skills or number identification,
thus obscuring the potential of these findings to
provide insight into early indicators of later difficulty or directions for early intervention. Similarly,
the spatial-attention pathway represented several
domain-general competencies, even though previous research suggests that domain-general competencies contribute differentially to various outcomes
(Hecht et al., 2003; Jordan et al., 2013; Seethaler
et al., 2011). For example, Jordan et al. (2013) found
that executive control shared a unique association
with fraction procedures but not fraction concepts,
whereas language and attentive behavior shared
unique associations with fraction concepts but not
fraction procedures. Second, Hecht et al. (2003)
found that domain-general competencies may not
be robust predictors of fraction knowledge. This
suggests the need to consider intervening specific
numerical precursor skills in the developmental
sequence of fraction learning.
Study Purpose and Overview
This longitudinal study examined a model of
fraction development that considered how firstand second-grade domain-general and numerical
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Vukovic et al.
competencies predict individual differences in fraction concepts at the end of fourth grade. We
targeted fourth grade because wide individual differences in fraction knowledge exist at that time,
even though formal fraction instruction only begins
in fourth grade. This signals the need to identify
early precursors of later difficulty so that schools
can identify children at risk of poor fraction development and provide targeted intervention using
strategies that align with the competencies involved
in the conceptual leap from whole number to fractions.
We initiated the study at the start of first grade,
when the range of individual differences on early
domain-general and numerical competencies is
already wide, based on informal learning experiences, and because the role of first-grade competencies has been established for the whole-number
mathematics learning that occurs in the primary
grades, prior to the introduction of fractions (e.g.,
Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant,
2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; Geary, 2011; Geary et al., 2009).
It is worth noting that we do not intend to imply
that domain-general competencies and their relation
with achievement—mathematics or otherwise—are
fixed at first grade. Instead, we intended to examine the potential of domain-general competencies to
serve as risk indicators of later fraction difficulty.
Domain-general competencies tend to become stable around first grade, and changes in individual
differences in these competencies after first grade
do not tend to alter the nature of their association
with mathematical development (e.g., Li & Geary,
2013; Sameroff, Seifer, Baldwin, & Baldwin, 1993).
Building on prior research, we studied five
domain-general competencies. First, nonverbal reasoning is hypothesized to play a role in the conceptual leap from whole number to fractions because
fraction knowledge is not intuitive and, compared
to whole numbers, involves greater symbolic complexity (Seethaler et al., 2011), which invokes the
type of mental flexibility, manipulation of symbolic
associations, and maintenance of multiple representations reflected in measures of nonverbal reasoning.
In a similar way, language skills are needed to operate in symbolically complex domains, and linguistic
skills play a vital role in the way children understand and represent mathematical concepts (e.g.,
LeFevre et al., 2010; Vukovic & Lesaux, 2013),
including fraction concepts (Jordan et al., 2013; Miura, Okamoto, Vlahovic-Stetic, Kim, & Han, 1999).
Third, attentive behavior is required to profit from
classroom instruction and to complete fraction tasks
that involve a complex sequence of steps (e.g., comparing fractions with different numerators and
denominators; Hecht et al., 2003). In such tasks,
executive control is also thought to play a supporting role by helping students hold the sequential set
of decisions in mind while completing subsequent
steps of the problem-solving process (Siegler &
Pyke, 2013). Finally, visual-spatial memory, which is
specialized for the maintenance of visual-spatial
information in immediate awareness, may play a
role in the conceptual leap because successful completion of fraction problems demands that children
store visual-spatial representations in mind while
manipulating these representations to solve fraction
problems (LeFevre et al., 2010). Note that whereas
the spatial-attention pathway in LeFevre et al.
(2010) represented several domain-general competencies, this study considered the independent
effects of nonverbal reasoning, attentive classroom
behavior, executive control, and visual-spatial memory, consistent with other researchers (Hecht et al.,
2003; Seethaler et al., 2011).
We also considered specific numerical competencies in our model. In line with previous researchers
(e.g., Geary, 2011; LeFevre et al., 2010), we hypothesized that number knowledge (e.g., representing
and manipulating exact quantities < 10) represents
a unique pathway to later fraction learning, separate from domain-general competencies. Number
knowledge is thought to be foundational to fraction
learning by providing the building blocks for
understanding abstract concepts relating to number,
magnitude, and proportional reasoning.
We proposed two types of intervening mathematical skills—in this case whole-number computations and number line estimation—in the
developmental progression from the basic firstgrade competencies to relatively sophisticated
fourth-grade fraction concepts. The first, competence with whole-number arithmetic, may interfere
with fraction development due to whole-number
bias (e.g., Ni & Zhou, 2005; Siegler & Pyke, 2013;
Stafylidou & Vosniadou, 2004). Indeed, Hecht et al.
(2003) found that fluency with whole-number combinations was not especially predictive of fraction
skill. However, Siegler and Pyke (2013) demonstrated that unlike low-achieving students, high
achievers were less susceptible to whole-number
bias, suggesting that mastery of whole-number
computations is foundational for fractions learning.
Given that instruction in the early school tends to
favor whole-number development, it is critical to
examine whether whole-number computations is a
precursor of fraction knowledge.
Understanding of Fractions
Second, we examined children’s rudimentary
proportional reasoning ability via a number line
estimation task. The ability to accurately place Arabic numerals on a number line with two endpoints
(e.g., 0 and 100) requires estimating a specific magnitude relative to the whole (Barth & Paladino,
2011; Slusser, Santiago, & Barth, 2013). Such proportion judgments—even with whole numbers—
may be foundational for later fraction knowledge,
given that both tasks require an understanding of
proportional magnitudes.
We measured whole-number computations and
number line estimation in second grade because
wide individual differences exist in both skills by
this time (e.g., Geary et al., 2009; Siegler & Booth,
2004). Furthermore, measuring these skills at a separate time point than the first-grade competencies
allowed for an examination of a fundamental developmental question surrounding the relative contribution of whole-number and proportional reasoning
to fraction learning and the interplay between these
skills and early competencies.
Specifically, the literature provides surprisingly
little insight into: (a) whether fraction acquisition is
an extension of the same learning pathway stemming from early cognitive competencies and wholenumber skills, (b) whether fraction learning follows
a learning trajectory from early cognitive competencies and proportional reasoning, (c) whether fraction
learning develops independent of both whole-number computations and proportional reasoning, or (d)
some combination thereof. Thus, this study sought
to assess the extent to which fraction concepts was
an extension of earlier forms of mathematical learning or if the conceptual leap required in fraction
learning relies predominantly on early domain-general or numerical competencies. Beginning to unravel the developmental course of fraction learning is
critical for informing instruction while generating
hypotheses for parallel conceptual leaps that occur
in the intermediate grades and for other forms of
learning at other developmental stages.
Method
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original project to fraction outcomes at end of the
fourth grade. The analytic sample was restricted to
the 163 children with complete data in first, second,
and fourth grades who also did not receive any
special intervention as part of the larger project
(because intervention was designed to disrupt the
predictive value of important variables). In the
larger project, an additional 80 students (~16
children/year; 6.6% annual attrition) would have
qualified for participation, but they had moved and
thus could not be assessed in fourth grade. Students in the present sample were not significantly
different from the 80 children who exited the study
on any of the first-grade predictor variables.
The sample’s mean age in first grade was 6 years
6 months (SD = 4 months); in second grade,
7 years 11 months (SD = 4 months); and in fourth
grade, 10 years 0 months (SD = 4 months). About
half of the children were female (n = 76 girls,
46.6%). According to school records, 76.7% of the
sample received subsidized lunch; 55.2% were
Black, 21.5% White non-Hispanic origin, 13.5%
White Hispanic origin, and 9.8% fell into other categories or information was not provided. There were
no sex differences on any of study measures, but
the White non-Hispanic group outperformed the
other groups on all study measures, as did the students who did not receive subsidized lunch. Given
these differences, racial/ethnic identity and lunch
status were controlled in all analyses. Note that all
children received the same curriculum.
Measures of First-Grade Predictors
Nonverbal Reasoning
The matrix reasoning subtest of the Wechsler
Abbreviated Scale of Intelligence (WASI; Wechsler,
1999) measures nonverbal reasoning with pattern
completion, classification, analogy, and serial reasoning tasks. Children complete an incomplete gridded pattern by pointing to or stating the number of
the correct response from five possible choices. As
reported in the test manual, reliability is between
.89 and .94 for first grade.
Participants
The data were collected with a sample of children in a Southeastern metropolitan school district
who participated in a larger research project
designed to examine the first-grade predictors of
emerging competence with whole-number arithmetic (Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010). We expanded the scope of the
Language
We considered two indicators of language ability: vocabulary and listening comprehension. With
the vocabulary subtest from the WASI (Wechsler,
1999), children orally define a series of words.
Responses are awarded a score of 0, 1, or 2 depending on conceptual quality. As reported by the test
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Vukovic et al.
developer, split-half reliability is between .86 and
.92. With the listening comprehension test of the
Woodcock Diagnostic Reading Battery (WDRB;
Woodcock, 1997), children listen to a short passage
and supply the single word missing at the end of
the passage. Reliability, as reported by the test
developer, is .80. We created a single language ability latent composite for use in subsequent analyses
(see Preliminary Analyses below).
Attentive Behavior
We used the Strengths and Weaknesses of
ADHD Symptoms and Normal Behavior Scale
(SWAN; Swanson et al., 2004) to index attentive
behavior. The SWAN is an 18-item teacher-rating
scale that measures children’s attentive behavior
and activity levels. It samples items from the Diagnostic and Statistical Manual of Mental Disorders–IV
(American Psychiatric Association, 1994) criteria for
attention deficit hyperactivity disorder for inattention (9 items) and hyperactivity impulsivity (9
items). Teachers rate each item on a 7-point scale
ranging from 1 (far below average) to 7 (far above
average). The reported score represents the average
rating across the 9 attentive behavior items. SWAN
correlates well with other dimensional assessments
of behavior related to attention (www.adhd.net).
Alpha in this study was .98.
Executive Control
We measured executive control with a task that
did not involve numbers to avoid artificially inflating the relation between executive control and mathematical outcomes based on superficial task features
(see Raghubar, Barnes, & Hecht, 2010). With the
listening recall subtest from the Working Memory
Test Battery for Children (WMTB–C; Pickering &
Gathercole, 2001), children hear a series of sentences
and first determine if each sentence is true (storage
demand); they then recall the last word in each sentence (processing demand). There are six items at
span levels from 1 to 6. Passing four items at a span
level moves the child to the next span level. At each
span level, the number of items to be remembered
increases by one. Failing three items terminates the
subtest. We report the trials correct score. As per the
test developer, test–retest reliability is .83.
Visual-Spatial Memory
We assessed visual-spatial memory with two
tasks from the WMTB–C (Pickering & Gathercole,
2001). With mazes memory, the tester presents a
maze with more than one solution and a picture of
an identical maze with a path showing one solution. The picture is removed after 3 s; then the
child duplicates the path. Note that the time constraint distinguishes this task from a spatial reasoning task (McGrew, 2005). Block recall uses a board
with nine raised blocks, each with a number on
one side, which only the tester sees. The tester taps
a block (or series of blocks); the child duplicates
the tapping. Passing four items at a level moves
the child to the next level. At each span level, the
number of items to be remembered increases by
one. Failing three items terminates the subtest. As
per the test developer, test–retest reliability is
.63–.68. Given the correlation between subtests
(r = .41, p < .001), we report the mean score across
both subtests.
Number Knowledge
The number sets test (Geary et al., 2009) indexes
the speed and accuracy with which children understand and operate with whole numbers < 10, while
partitioning sets and transcoding between quantities (sets of objects, such as circles and stars) and
symbols (Arabic numerals). Children are presented
with two pages of domino-like stimuli in which
the squares of the domino are objects (e.g., circles,
stars), Arabic numerals, or blank. At the top of the
page, the target sum (5 or 9) is shown. The child
is instructed to “circle any groups that can be put
together to make the top number” and “work as
fast as you can without making many mistakes.”
For example, on the first page, the child circles
any domino in which all squares (mostly twosquare dominos but also some three-square dominos), when added, result in 5 (e.g., domino with
two circles and three stars). The student has 60 s
for the target sum 5; 90 s for the target sum 9. Signal detection methods are applied to the number
of hits and false alarms to generate a d’ variable
representing sensitivity to quantities (Geary et al.,
2009). Children who correctly identify many target
quantities and commit few false alarms have high
scores. Individual differences in d’ are likely
related to (a) acuity of the cognitive systems for
representing exact small quantities or approximate
larger ones (Koontz & Berch, 1996), (b) implicit or
explicit understanding of addition (Levine, Jordan,
& Huttenlocher, 1992), and (c) understanding of
cardinal value and that numbers are composed of
sets of smaller magnitude numbers (Butterworth,
2010).
Understanding of Fractions
Measures of Second-Grade Intermediate Numeracy Skills
Number Line Estimation
With the number line estimation task (Siegler &
Booth, 2004), children estimate where whole numbers
fall on a number line. Stimuli are 24 number lines
displayed across the center of a standard computer
screen. Each number line is 25 cm and has a start
point of 0 and an endpoint of 100 with a target number printed approximately 5 cm above it in a large
font (72 pt). Target numbers are 3, 4, 6, 8, 12, 17, 21,
23, 25, 29, 33, 39, 43, 48, 52, 57, 61, 64, 72, 79, 81, 84,
90, and 96. Stimuli are presented in random order for
each child. The tester first explains a number line that
includes the 0 and 100 endpoints and is marked in
increments of 10. After the tester judges that the child
recognizes the concept, a number line containing
only the endpoints 0 and 100 is presented, and the
child points to where 50 should go. A model number
line with the endpoints and the location of 50 marked
is then presented, and the child compares his or her
response to the model. The tester explains how “the
number 50 is half of 100, so we put it halfway in
between 0 and 100 on the number line.” Next, the
tester teaches the child to use the arrow keys to place
a red pointer on the line where 50 should fall on the
computer screen. Then, the measure is administered.
For each item, the tester says, “If this is zero (pointing) and this is 100 (pointing), where would you put
N?” There is no time constraint. We used the absolute
value of the difference between the correct placement
and the child’s placement (i.e., estimation of accuracy), averaged across trials; smaller values indicate
greater accuracy. This accuracy score correlates with
mathematics achievement (Siegler & Booth, 2004).
Cronbach’s alpha on the sample was .93.
Arithmetic Computation
With the Arithmetic subset of the Wide Range
Achievement Test–3 (WRAT–3; Wilkinson, 1993),
children complete written calculation problems of
increasing difficulty that are presented in horizontal
or vertical format. Items are not read to the children.
Most of this second-grade sample completed wholenumber addition and subtraction problems with single and double digits, with and without regrouping;
some children also completed simple multiplication
problems. In the present sample, alpha was .87.
Fourth-Grade Conceptual Understanding of Fractions
Understanding of fraction concepts was assessed
using 17 released fraction items from recent
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National Assessments of Educational Progress
(NAEP; National Center for Education Statistics,
2007, 2009). With the NAEP items, testers read each
problem aloud (with up to one rereading upon student request). Eight items assess the part–whole
interpretation of fractions (e.g., “Shade 3/8 of the
rectangle above”), eight assess the measurement
interpretation of fractions (e.g., “Which picture
shows that 2/3 is the same as 6/9?”), and one asks
how many fourths make a whole. Students select
an answer from four choices (10 problems), write
an answer (3 problems), shade a portion of a fraction (1 problem), mark a number line (1 problem),
write a short explanation (1 problem), or write
numbers, shade fractions, and explain the answer
(1 problem with multiple parts). Alpha was .73.
Control Variables
All analyses included controls for the demographic characteristics noted and general achievement levels in first grade. To index general
academic ability, we used the reading subtest of the
WRAT–3 (Wilkinson, 1993) consistent with previous
research (Siegler & Pyke, 2013). Children read
words of increasing difficulty. Alpha is .94 at age 6.
To index general mathematics ability, we used the
concepts and applications test (Fuchs, Hamlett, &
Fuchs, 1990), consistent with previous research
(Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant,
2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010). With 25 items, this test samples
numeration, concepts, geometry, measurement,
applied computation, money, charts, and word
problems. For example, items require counting
arrays of items, writing the number after, identifying smaller versus larger quantities, and identifying
operators within number sentences. Alpha was .93.
Procedure
At the beginning of first grade, children were
assessed on nonverbal reasoning, language, attentive behavior, executive control, visual-spatial memory, and number knowledge. In spring of second
grade, children were assessed on number line estimation and whole-number computations. In spring
of fourth grade, children were assessed on the
NAEP fraction items. Testing occurred individually
in a quiet room for all tasks except for number
knowledge in first grade and NAEP items in fourth
grade, which were group administered. Graduate
student testers were trained to criterion at each testing occasion and used standard directions for
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administration. Reliability across testers was
assessed on percentage of agreement in scoring,
based on audiotapes of children’s responses; agreement exceeded 99%.
Data Analytic Plan
Mediation analysis following Baron and Kenny
(1986) and using the Preacher and Hayes (2008)
SPSS macro was used to examine the extent to
which fraction concepts were an extension of earlier
forms of mathematical learning or if the conceptual
leap required in fraction learning relied predominantly on early domain-general or numerical competencies. Mediation analysis has the advantage of
allowing researchers to investigate direct associations, which in this study was used to examine the
relation between first-grade competencies and
fourth-grade fraction concepts, while holding constant the intermediate mathematical skills (and control variables). Importantly, mediation analysis also
provides estimations of the statistical significance of
indirect associations, which was used in this study
to evaluate whether whole-number computations
and proportional reasoning interrupted the association between first-grade competencies and fourthgrade fraction concepts. Bootstrapping (5,000 draws
to estimate standard errors) was used to construct
95% confidence intervals for the indirect associations. Bootstrapping is particularly valuable for estimating indirect associations because it makes fewer
assumptions than other tests of indirect effects (e.g.,
shape of sampling distribution and standard error
of indirect effect), for which traditional z statistics
are known to be biased (Fiedler, Schott, & Meiser,
2011; Gelfand, Mensinger, & Tenhave, 2009; Hayes,
2009; MacKinnon, Fairchild, & Fritz, 2007). We
remind readers that because mediation analyses are
correlational, causation should not be inferred.
Results
Preliminary Analyses
See Table 1 for sample means in standard scores
(as available) as well as raw scores, which were
used in subsequent analyses. As the standard scores
indicate, the sample performed in the average range
on all standardized measures. Table 1 also shows
correlations among all first-grade competencies, second-grade mathematical skills, and fourth-grade
fraction concepts. The first-grade competencies were
moderately to strongly correlated with the secondgrade mathematics skills and fourth-grade fraction
concepts. The second-grade mathematics skills were
also moderately to strongly correlated with fourthgrade fraction concepts.
Exploratory analyses were conducted to investigate the appropriateness of forming a latent language composite based on the vocabulary and
listening comprehension variables. One factor was
extracted, which accounted for 81.23% of the variance and was well defined by both variables (standardized loadings = .90). We used this latent
composite in subsequent analyses.
Mediation Analyses: Unraveling the Developmental
Course of Fraction Learning
The primary analyses focused on examining the
direct and indirect associations between the firstand second-grade variables and fourth-grade fraction concepts. All analyses included controls for
beginning of first-grade general mathematics
achievement and reading achievement, demographic variables (i.e., racial/ethnic identity and
subsidized lunch status), and all first-grade competencies. For instance, when testing the specific association of nonverbal reasoning, we controlled for
language, attentive behavior, executive control,
visual-spatial memory, number knowledge, concepts and applications, reading, racial/ethnic identity, and subsidized lunch status. Results are
summarized in Figure 1.
As shown at the top of Table 2, the association of
both second-grade mathematics skills with fourthgrade fraction concepts—b paths—was statistically
significant, holding constant first-grade competencies and controls. Thus, second-grade mathematical
skills met criteria for potential mediation between
first-grade competencies and fourth-grade fraction
concepts. The middle panel of Table 2—a paths—
shows that first-grade language was the only
variable that was significantly associated with both
second-grade number line estimation and secondgrade whole-number computations, after controlling
for all control variables. First-grade visual-spatial
memory had a significant association with secondgrade number line estimation, whereas first-grade
attentive behavior had a significant association with
second-grade whole-number computations. Firstgrade nonverbal reasoning, executive control, and
number knowledge did not have significant
associations with second-grade number line estimation or second-grade whole-number computations,
after accounting for control variables. First-grade
number knowledge had a direct association with
fourth-grade fraction concepts.
Understanding of Fractions
1469
Table 1
Means Raw Scores, Mean Standard Scores, and Correlations Among First-, Second- and Fourth-Grade Variables
Raw score
M (SD)
1. G1 reading
2. G1 concepts &
applications
3. G1 nonverbal
reasoning
4. G1 language
composite
5. G1 vocabulary
6. G1 listening
comprehension
7. G1 attentive
behavior
8. G1 listening recall
9. G1 visual-spatial
memory
10. G1 number
knowledge
11. G2 number
line estimation
12. G2 arithmetic
computations
13. G4 NAEP
fractions
Standard score
M (SD)
Correlations
1
2
—
.69
—
.43
.52
—
.58
.60
.39
—
.53
.52
.49
.60
.39
.31
39.06 (12.60)
.55
.66
5.40 0(4.11)
11.26 0(3.34)
.58
.40
0.21 0(1.04)
12.93 0(6.59)
20.61 0(4.94)
11.22 0(5.16)
105.12 (15.32)
8.71 0(5.23)
48.94 0(9.26)
0.00 0(1.00)
17.28 0(6.76)
14.08 0(5.18)
21.47 0(3.31)
41.24 (10.68)
87.39 (16.55)
94.95 (13.50)
11.79 0(4.24)
3
4
5
6
7
8
9
.90
.90
—
.63
—
.42
.50
.44
.45
—
.64
.45
.45
.46
.65
.35
.59
.32
.58
.31
.49
.66
.48
.50
.46
.43
.53
.33
.47
.50
.53
.38
.56
.68
.49
10
11
12
.54
.39
—
.47
—
.44
.58
.53
.43
—
.43
.41
.34
.36
.38
.43
—
.50
.51
.39
.52
.48
.40
.50
.46
—
.53
.48
.48
.57
.50
.41
.67
.52
.58
Note. G1 = Grade 1; G2 = Grade 2; G4 = Grade 4; NAEP = National Assessments of Educational Progress. Standardized scores for
nonverbal reasoning and vocabulary represent T-score values (M = 50, SD = 10); standardized scores for listening comprehension represent standard scores (M = 100, SD = 15). All correlations had p values < .001.
G1 Nonverbal Reasoning
Path a = -1.68
Language Path ab = .14
G1 Language
G2 Number Line
Estimation
Path a = .60
G1 Attentive Behavior
Visual-Spatial Memory Path ab = .03
Path b = -.08
Path a = .05
Path b = .21
G1 Executive Control
G2 Arithmetic
Computations
G1 Visual-Spatial Memory
G4 Fraction Concepts
Attentive Behavior Path ab = .01
Language Path ab = .12
Path a = -.34
G1 Number Knowledge
Path c1 = 1.13
Figure 1. Direct and indirect associations between first-grade competencies (left), second-grade intervening mathematical skills (middle),
and fourth-grade fraction concepts (right). Analyses controlled for demographic characteristics, general mathematics ability, and general
academic ability. Solid black lines indicate significant direct associations. Broken gray lines indicate significant indirect associations.
Nonsignificant paths are not displayed. Note that paths are negative for number line estimates because lower scores indicate greater
accuracy. G1 = Grade 1; G2 = Grade 2; G4 = Grade 4.
Table 3 displays the total and indirect associations of the first-grade competencies on fourthgrade fraction concepts; 95% confidence intervals
that do not cover 0 are statistically significant.
First-grade language had a significant indirect
association with fourth-grade fraction concepts via
1470
Vukovic et al.
Table 2
Coefficients for Path a, Path b, and Path c1 in Mediation Models
Path
Coefficient
(SE)
t value
Table 3
Indirect Effects of First-Grade Predictors on Fourth-Grade Fraction
Concepts
p value
Path
Path b: Effect of Grade 2 skills
on Grade 4 fractions concepts
Number line estimation ?
.083 (.04)
1.98
.050
Fraction concepts
Arithmetic computations ?
.206 (.09)
2.35
.020
Fraction concepts
Path a: Effect of Grade 1
predictor on Grade 2 skill
Nonverbal reasoning ?
.061 (.11)
0.56
.578
Number line estimation
Nonverbal reasoning ?
.008 (.05)
0.15
.884
Arithmetic computations
Language ability ?
1.675 (.62)
2.69
.008
Number line estimation
Language ability ?
.600 (.30)
2.00
.047
Arithmetic computations
Attentive behavior ?
.047 (.05)
0.97
.336
Number line estimation
Attentive behavior ?
.054 (.02)
2.33
.021
Arithmetic computations
Executive control ?
.207 (.16)
1.31
.193
Number line estimation
Executive control ?
.021 (.08)
0.27
.787
Arithmetic computations
Visual-spatial memory ?
.340 (.16)
2.17
.032
Number line estimation
Visual-spatial memory ?
.019 (.08)
1.44
.153
Arithmetic computations
Number knowledge ?
.581 (.61)
0.95
.344
Number line estimation
Number knowledge ?
.313 (.29)
1.06
.290
Arithmetic computations
Path c1: Direct effect of Grade 1 predictor on Grade 4 fraction
concepts
Nonverbal reasoning ?
.071 (.06)
1.29
.200
Fractions concepts
Language ability ?
.218 (.32)
0.67
.503
Fractions concepts
Attentive behavior ?
.025 (.02)
1.01
.315
Fractions concepts
Executive control ?
.052 (.08)
0.64
.521
Fractions concepts
Visual-spatial memory ?
.006 (.08)
0.07
.943
Fractions concepts
Number knowledge ?
1.128 (.31)
3.63
.000
Fractions concepts
Note. In all models, the effects of the other first-grade predictor
variables were controlled, as were the effects of Grade 1 reading
and math ability, racial/ethnic, and subsidized lunch status.
both second-grade number line estimation and
whole-number computations. First-grade attentive
behavior had a statistically significant indirect
Total effect of nonverbal
reasoning
Via Grade 2 number line
estimation
Via Grade 2 arithmetic
computations
Total effect of
language ability
Via Grade 2 number
line estimation
Via Grade 2 arithmetic
computations
Total effect of
attentive behavior
Via Grade 2 number
line estimation
Via Grade 2 arithmetic
computations
Total effect of executive
control
Via Grade 2 number
line estimation
Via Grade 2 arithmetic
computations
Total effect of
visual-spatial memory
Via Grade 2 number
line estimation
Via Grade 2 arithmetic
computations
Total effect of number
knowledge
Via Grade 2 number
line estimation
Via Grade 2 arithmetic
computations
Indirect effect
path (ab)
Bootstrapped
95% CI
.067 (.06)
.0088 to .0125
.0004 (.00)
.0061 to .0076
.0010 (.00)
.0046 to .0086
.2592 (.11)
.0846 to .5335
.1399 (.08)
.0218 to .3598
.1193 (.08)
.0106 to .3306
.0069 (.01)
.0076 to .0279
.0037 (.00)
.0162 to .0028
.0106 (.01)
.0011 to .0304
.0147 (.02)
.0626 to .0367
.0180 (.02)
.0633 to .0056
.0033 (.02)
.0235 to .0429
.0498 (.03)
.0124 to .1226
.0280 (.02)
.0034 to .0779
.0218 (.02)
.0007 to .0788
.1166 (.11)
.0558 to .3829
.0522 (.06)
.0383 to .2234
.0644 (.07)
.0268 to .2821
Note. In all models, the effects of the other first-grade predictor
variables were controlled, as were the effects of Grade 1 reading
ability, Grade 1 general math ability, racial/ethnic, and subsidized lunch status. Confidence intervals (CIs) that do not cover 0
are statistically significant.
association via second-grade whole-number computations. First-grade visual-spatial memory had a
statistically significant indirect association via second-grade number line estimation; the indirect
association via second-grade whole-number computation appeared significant but could not be considered as a mediator given the nonsignificant a
path from visual-spatial memory to whole-number
computation.
Understanding of Fractions
Discussion
The goal of this study was to begin to unravel the
developmental course of fraction learning, especially whether conceptual understanding of fractions is an extension of earlier forms of
mathematical learning or if the conceptual leap
from whole-number understanding to fraction
knowledge is predicated on other forms of learning.
At this time, the developmental precursors of fraction knowledge are poorly understood, which is
especially problematic considering the central role
fractions play in economic and social opportunities
in adulthood (National Mathematics Advisory
Panel, 2008; Siegler et al., 2012). We tested the relative contributions of domain-general and numerical
competencies to the prediction of fraction concepts.
Overall, results suggest that domain-general and
numerical competencies play different roles in the
conceptual leap from whole numbers to fractions,
findings that may also be relevant for understanding conceptual leaps occurring in other academic
domains or at other developmental stages.
Domain-General Competencies Support Building Blocks
for Later Learning
The associations between domain-general competencies and children’s later understanding of fraction concepts were all mediated by second-grade
mathematical skills. Indeed, the findings suggest
that domain-general competencies—language, visualspatial memory, and attentive behavior in particular—at school entry facilitate the learning of
mathematical skills that provide the building blocks
for later fractions learning, but are not direct antecedents of fraction learning. These findings lend
strength to the idea that learning fraction concepts
is an extension of earlier forms of mathematical
learning, and that domain-general competencies
play a secondary role to mathematical skills. This
does not mean that domain-general competencies
assessed simultaneously with fractions concepts
would not be important, but does indicate that
early competencies operate through intermediate
mathematical skills.
That language was associated with intervening
mathematical skills but was not a direct antecedent
of fraction concepts contrasts with the hypothesis
that linguistic skills influence the kinds of mathematical learning that depend on the symbolic number system, including fractions (LeFevre et al.,
2010). Whereas LeFevre et al. (2010) found that
their linguistic composite measured at 4.5 years of
1471
age predicted all mathematical outcomes (i.e.,
whole-number computations, numeration, geometry, measurement, and number line estimation)
3 years later, a more complex association emerged
in this study, with language skills predicting subsequent whole-number computations and number
line estimation, which in turn predicted later fraction concepts. These findings thus suggest that
early language competencies play a more nuanced
role in fraction development than proposed by
LeFevre et al. It is possible that early language
competencies promote the development of intervening mathematical skills rather than directly support
how children understand fraction concepts. For
example, language capability helps children cognitively represent large numbers (i.e., ≥ 5) with precision, which is needed to manipulate exact
quantities, as with whole-number computations
(e.g., 5 + 9), and to estimate exact magnitudes relative to each other, as in making number line estimations (e.g., accurately placing 5 on a number line).
Early language competency may therefore be
needed to master second-grade mathematical skills,
which in turn impart an advantage in learning
more nuanced features of number necessary for
fourth-grade fraction knowledge, such as understanding the difference between 5 and 1/5. Thus,
poor early language competency may forecast children’s difficulty making the conceptual leap from
whole number to fractions, signaling the need to
ensure mastery of mathematical language in the
early years to provide a solid foundation for
learning both whole numbers and fractions.
In this study, visual-spatial memory and attentive behavior were associated with fraction concepts, whereas executive control and nonverbal
reasoning were not. This finding thus suggests that
not all domain-general competencies associated
with the spatial-attention pathway proposed by
LeFevre et al. (2010) are associated with mathematical learning—fractions in this case—a finding that
would have been obscured had we considered a
composite spatial-attention variable. Early visualspatial competencies may reflect the ease with
which children learn to (a) map spatial representations of magnitude onto precise symbolic representations (by second grade, children have developed
mappings between nonverbal magnitude representations and Arabic numerals up to around 100),
and (b) hold in memory spatial and symbolic representations of magnitude to make proportion judgments (e.g., accurately placing 5 on a 0–100 number
line). Early visual-spatial memory may thus be
needed to integrate the visual-spatial and symbolic
1472
Vukovic et al.
representations necessary to make accurate number
line estimations, which may in turn assist children
in interpreting visual-spatial and symbolic connections related to fourth-grade fractions, such as producing a visual representation of 1/5. If so, the
present findings would suggest that children with
weak visual-spatial memory may have difficulty
making the transition to fractions unless they
receive early support in learning to integrate visualspatial and symbolic representations of magnitude,
as in making number line estimations.
Attentive behavior in this study was associated
with whole-number computations, consistent with a
substantial body of research (e.g., Fuchs, Geary,
Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs,
Geary, Compton, Fuchs, Hamlett, Seethaler, et al.,
2010), but not with number line estimation or fraction concepts, in contrast to previous findings
(Hecht et al., 2003; Jordan et al., 2013). Whereas
Hecht et al. (2003) and Jordan et al. (2013) investigated this association in third through fifth graders,
this study focused on younger children. This may
mean that in the early school years, attentive
behavior matters specifically for the acquisition of
procedural skills, whereas conceptual development
depends less on attentive behavior. Attentive
behavior may help children master the sequenced
procedures needed to solve arithmetic problems,
which becomes especially important when children
learn regrouping in second grade. This early attentive ability needed to master second-grade mathematical rules and procedures may in turn support
the conceptual leap to fractions in fourth grade,
when children must distinguish between the rules
and procedures related to whole numbers versus
fractions (e.g., multiplying whole numbers always
results in a larger number, whereas multiplying
fractions does not). These findings suggest that
poor attentional capacity is a risk factor for later
fraction learning. Teachers may need to be intentional in directing and ensuring children’s attention
during mathematical instruction in ways that focus
on rules and procedures, especially for students
who exhibit poor attentive behavior.
It must be noted that the findings as they relate
to teasing apart the roles of executive control and
visual-spatial memory are exploratory and our
interpretations thus speculative. Researchers have
noted the problem of task impurity with measures
of executive processes, whereby tasks designed to
measure specific executive functions often place
demands on several executive processes and/or
nonexecutive processes (e.g., Best, Miller, & Jones,
2009; Miyake et al., 2000), making it difficult to
interpret findings pertaining to any one measure.
One solution has been to use a latent variable
approach to capitalize on the underlying competencies rather than surface features of the task, an
approach not possible in this study. Although our
results are consistent with the finding that visualspatial memory has broader effects on mathematical
learning than does executive control (Bull, Espy, &
Wiebe, 2008; St Clair-Thompson & Gathercole,
2006), and that attentive behavior is important for
learning whole-number procedures (e.g., Fuchs,
Geary, Compton, Fuchs, Hamlett, & Bryant, 2010;
Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler,
et al., 2010; Fuchs et al., 2006), this study should be
replicated using multiple measures of executive
processes.
Multiple Pathways to Fraction Competence
The second main finding was that fraction concepts had roots in earlier forms of mathematical
learning and in early numerical competencies. This
lends support to the idea that not only is learning
fractions an extension of earlier forms of mathematical learning but also that there may be multiple
pathways to fraction competence. Specifically, the
findings suggest that children may learn fractions
through (a) the same learning pathway as wholenumber development, (b) a proportional reasoning
learning trajectory, or (c) a yet unspecified developmental course stemming from early numerical
competencies.
In this study, number knowledge was the only
first-grade competency that directly predicted
fourth-grade fraction concepts. This corroborates
LeFevre et al. (2010) and Butterworth (2010) who
proposed a quantitative pathway as a unique source
of individual differences in mathematical skill independent of domain-general competencies. Interestingly, first-grade number knowledge was not
associated with second-grade arithmetic computations or number line estimation, a somewhat surprising finding given that both second-grade tasks
required numerical processing. Thus, the findings
suggest a unique pathway to fraction learning that is
independent from whole-number computations and
proportional reasoning. Butterworth and colleagues
posit that the ability to recognize, represent, and
manipulate small numerosities provides the building
blocks for learning arithmetic in particular (e.g., Butterworth, 2010; Gelman & Butterworth, 2005). By
contrast, our results suggest that the fluency with
which children can identify, process, and manipulate
small quantities, as assessed with the number
Understanding of Fractions
sets test (Geary et al., 2009), matters specifically for
fraction learning. This fluency may serve as a proxy
for children’s later ability to make the conceptual
leap from whole-number understanding to fraction
understanding, although the underlying mechanism
remains to be identified. In any case, results suggest
that the number sets test, a highly efficient measure,
may be diagnostically useful in forecasting children’s
difficulty with fraction concepts, thereby providing a
means of identifying students for early intervention.
It also suggests that an emphasis on number knowledge and fluency in recognizing and manipulating
small quantities may be a target for early intervention to prevent fraction difficulty.
That second-grade number line estimation was
uniquely associated with fraction concepts 2 years
later suggests that children’s rudimentary proportional reasoning ability also provides a foundation
for learning fractions. This finding corroborates Jordan et al. (2013) with a different sample spanning a
longer developmental period and while considering
mediating effects. In Jordan et al., third-grade number line estimation made the largest unique contribution to fourth-grade fraction concepts above the
effects of third-grade whole-number skills, language, and attentive behavior. Together, results
indicate that children—even those as young as second grade—who make accurate number line estimations seem also to have an advantage in learning
fraction concepts. Given the increased emphasis on
the number line in the Common Core State Standards, this finding is likely relevant to practitioners.
Results also suggest that proportional reasoning
might have a different learning trajectory separate
from whole-number concepts and procedures. This
corroborates Mix et al.’s (1999) finding that children
have an informal understanding of fractions before
formal fraction instruction occurs. This study identified two developmental precursors of number line
estimation, namely, language and visual-spatial
memory. Future research is needed to further specify this learning trajectory, especially the extent to
which nonsymbolic proportional reasoning (e.g.,
Spence & Krizel, 1994) and the approximate number system (ANS; Halberda, Mazzocco, & Feigenson, 2008; Libertus, Feigenson, & Halberda, 2011)
are involved. Although Jordan et al. (2013) found
that the ANS in third grade did not uniquely predict fourth-grade fraction outcomes, few studies
have considered how nonsymbolic proportional
reasoning contributes to the development of fraction knowledge independent of whole-number competencies or how the ANS contributes to number
line development in the early school years.
1473
Finally, results suggest that whole-number skills
are developmental precursors of fraction learning.
Hecht et al. (2003) found that whole-number arithmetic computations were predictive of procedural
fraction skill. This study extends these findings to
indicate that whole-number computations are also
predictive of children’s understanding of fraction
concepts. Although the iterative relation between
conceptual and procedural knowledge has been
well documented (e.g., Rittle-Johnson & Alibali,
1999; Rittle-Johnson et al., 2001), a unique contribution of this study is the finding that whole-number
procedural skill predicted conceptual understanding
of fractions 2 years later. This finding is especially
striking given that general mathematics ability, general academic ability, and proportional reasoning
were controlled. This finding provides strong support for the hypothesis that the conceptual leap
from whole number to fractions depends in part on
mastery of whole-number concepts and procedures
(National Mathematics Advisory Panel, 2008; Siegler & Pyke, 2013). Thus, findings suggest that the
whole-number bias may be alleviated in part by
addressing gaps in children’s understanding of
whole-number procedures and principles.
Limitations and Conclusions
Readers should note that as with all studies,
specific measures were selected to operationalize
constructs, and varying the methods for indexing
key abilities or expanding the models to incorporate
additional competing predictors could alter findings. Also, this study did not examine all potential
competencies that might have an influence on children’s fraction development, most notably the ANS
(e.g., Halberda et al., 2008) and nonsymbolic proportional reasoning (e.g., Spence & Krizel, 1994).
Future research is needed to determine which
domain-general and numerical competencies are
involved in later fractions learning, including specifying the independent pathways identified in this
study. We further note that like other correlational
analyses, evidence of significant associations in our
model cannot be interpreted as causal nor can it be
assumed that the proposed model represents the
only or best solution. For instance, a significant
mediated association could reflect that the intermediate mathematical skill is a correlate of either the
true (untested) mediator or of fraction concepts and
not necessarily a causal mechanism. It is thus
important to replicate the present findings.
Finally, although difficulty with fractions is not
just a U.S. phenomenon (e.g., Chan et al., 2007;
1474
Vukovic et al.
Tatsuoka et al., 2004), it is interesting to consider
the role of school context in explaining the development of fraction knowledge. Specifically, the
observed findings may depend on the type of
instruction provided at that time in American
schools, where the dominant focus is part–whole
understanding. A second type of understanding,
the measurement interpretation of fractions, reflects
cardinal size. This type of interpretation, often represented with number lines, has until recently been
assigned a subordinate role, addressed later and
with less emphasis. However, an instructional focus
on the measurement interpretation is in keeping
with the Common Core State Standards’ explicit
emphasis on understanding of fraction equivalence
and ordering. A schooling context that emphasizes
the measurement interpretation may result in different findings. Future research is needed to tease
apart how differences in instruction, for example,
part–whole versus measurement interpretation,
facilitate or hinder fraction learning.
With these caveats in mind, we offer two conclusions. First, domain-general competencies may be
foundational for fraction concepts only insofar as
they promote the development of intermediate
mathematical skills. Not all domain-general competencies, however, are necessarily implicated. Language ability appears to matter generally for
mathematical development, whereas visual-spatial
memory matters for number line estimation, and
attentive behavior matters for arithmetic computations. Nonverbal reasoning and executive control
did not emerge as salient predictors of children’s
understanding of fraction concepts. Second, early
numerical and mathematical competencies may be
direct antecedents and powerful longitudinal
predictors of children’s understanding of fraction
concepts.
Competence with fractions plays a pivotal role in
determining success in and out of school. Our findings suggest that children develop fraction competence through multiple pathways, indicating that a
breakdown in learning fractions may occur at any
point along any particular pathway, beginning with
early domain-general and numerical competencies.
Just as reading comprehension difficulties may
result from decoding or language comprehension
deficits (among other sources), fraction difficulties
may stem from multiple origins that require different instructional interventions. For example,
children with adequate whole-number knowledge
may nonetheless experience difficulty learning fractions because of weaknesses in understanding number generally or in understanding proportional
reasoning. Such weaknesses might further reflect
underlying difficulties in domain-general functioning, such as language or visual-spatial memory, or
gaps in mathematical content related to number
and proportional reasoning. Each, in turn, would
have different instructional implications. Future
research is needed to specify the precise sequence
of learning that occurs along different trajectories to
inform diagnostic assessments and corresponding
instructional interventions.
References
American Psychiatric Association. (1994). Diagnostic and
statistical manual of mental disorders (4th ed.). Washington, DC: Author.
Baron, R. M., & Kenny, D. A. (1986). The moderator–
mediator variable distinction in social psychological
research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51,
1173–1182. doi:10.1037/0022-3514.51.6.1173
Barth, H. C., & Paladino, A. M. (2011). The development
of numerical estimation: Evidence against a representational shift. Developmental Science, 14, 15–135. doi:
10.1111/j.1467-7687.2010.00962.x
Best, J. R., Miller, P. H., & Jones, L. L. (2009). Executive
functions after age 5: Changes and correlates. Developmental Review, 29, 180–200. doi:10.1016/j.dr.2009.05.002
Bull, R., Espy, K. A., & Wiebe, S. A. (2008). Short-term
memory, working memory, and executive functioning
in preschoolers: Longitudinal predictors of mathematical achievement at age 7 years. Developmental Neuropsychology, 33, 205–228. doi:10.1080/87565640801982312
Butterworth, B. (2010). Foundational numerical capacities
and the origins of dyscalculia. Trends in Cognitive
Sciences, 14, 534–541. doi:10.1016/j.tics.2010.09.007
Calhoon, M. E., Emerson, M. B., Flores, M., & Houchins,
D. E. (2007). Computational fluency performance profile of high school students with mathematics disabilities. Remedial & Special Education, 28, 292–303. doi:
10.1177/07419325070280050401
Chan, W. H., Leu, Y. C., & Chen, C. M. (2007). Exploring
group-wise conceptual deficiencies of fractions for fifth
and sixth graders in Taiwan. Journal of Experimental
Education, 76, 26–57. doi:10.3200/JEXE.76.1.26-58
Fiedler, K., Schott, M., & Meiser, T. (2011). What mediation analysis can (not) do. Journal of Social Psychology,
47, 1231–1236. doi:10.1016/j.jesp.2011.05.007
Fuchs, L. S., Fuchs, D., Compton, D. L., Powell, S. R.,
Seethaler, P. M., Capizzi, A. M., … Fletcher, J. M.
(2006). The cognitive correlates of third-grade skill in
arithmetic, algorithmic computation, and arithmetic
word problems. Journal of Educational Psychology, 98,
29–43. doi:10.1037/0022-0663.98.1.29
Fuchs, L. S., Geary, D. C., Compton, D. L., Fuchs, D.,
Hamlett, C. L., & Bryant, J. V. (2010). The contributions
of numerosity and domain-general abilities to school
Understanding of Fractions
readiness. Child Development, 81, 1520–1533. doi:10.
1111/j.1467-8624.2010.01489.x
Fuchs, L. S., Geary, D. C., Compton, D. L., Fuchs, D.,
Hamlett, C. L., Seethaler, P. M., … Schatschneider, C.
(2010). Do different types of school mathematics development depend on different constellations of numerical
and general cognitive abilities? Developmental Psychology, 46, 1731–1746. doi:10.1037/a0020662
Fuchs, L. S., Hamlett, C. L., & Fuchs, D. (1990). Curriculum-based math computation and concepts/applications.
(Available from L. S. Fuchs, 328 Peabody, Vanderbilt
University, Nashville, TN 37203.)
Geary, D. C. (2011). Cognitive predictors of individual
differences in achievement growth in mathematics: A
five year longitudinal study. Developmental Psychology,
47, 1539–1552. doi:10.1037/a0025510
Geary, D. C., Bailey, D. H., & Hoard, M. K. (2009). Predicting mathematical achievement and mathematical
learning disability with a simple screening tool: The
number sets test. Journal of Psychoeducational Assessment,
27, 265–279. doi:10.1177/0734282908330592
Gelfand, L. A., Mensinger, J. L., & Tenhave, T. (2009).
Mediation analysis: A retrospective snapshot of practice
and more recent directions. Journal of General Psychology, 136, 153–176. doi:10.3200/GENP.136.2.153-178
Gelman, R., & Butterworth, B. (2005). Number and language: How are they related? Trends in Cognitive Sciences, 9, 6–10. doi:10.1016/j.tics.2004.11.004
Halberda, J., Mazzocco, M. M. M., & Feigenson, L. (2008).
Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 455, 665–668.
doi:10.1038/nature07246
Hayes, A. F. (2009). Beyond Baron and Kenny: Statistical
mediation analysis in the new millennium. Communication Monographs, 76, 408–420. doi:10.1080/0363775
0903310360
Hecht, S., Close, L., & Santisi, M. (2003). Sources of individual differences in fraction skills. Journal of Experimental Child Psychology, 86, 277–302. doi:10.1016/j.jecp.2003.
08.003
Jordan, N. C., Hansen, N., Fuchs, L. S., Siegler, R. S., Gersten, R., & Micklos, D. (2013). Developmental predictors of fraction concepts and procedures. Journal of
Experimental Child Psychology, doi:10.1016/j.jecp.2013.02.
001
Koontz, K. L., & Berch, D. B. (1996). Identifying simple
numerical stimuli: Processing inefficiencies exhibited by
arithmetic learning disabled children. Mathematical
Cognition, 2, 1–23. doi:10.1080/135467996387525
LeFevre, J., Fast, L., Skwarchuk, S., Smith-Chant, B. L.,
Bisanz, J., Kamawar, D., & Penner-Wilger, M. (2010).
Pathways to mathematics: Longitudinal predictors of
performance. Child Development, 81, 1753–1767. doi:
10.1111/j.1467-8624.2010.01508.x
Levine, S. C., Jordan, N. C., & Huttenlocher, J. (1992).
Development of calculation abilities in young children.
Journal of Experimental Child Psychology, 53, 72–103.
doi:10.1016/S0022-0965(05)80005-0
1475
Li, Y., & Geary, D. C. (2013). Developmental gains in
visuospatial memory predict gains in mathematics
achievement. PLoS ONE, 8, e70160. doi:10.1371/journal.
pone.0070160
Libertus, M. E., Feigenson, L., & Halberda, J. (2011).
Preschool acuity of the approximate number system
correlates with school math ability. Developmental Science, 14, 1292–1300. doi:10.1111/j.1467-7687.2011.01080.x
MacKinnon, D. P., Fairchild, A. J., & Fritz, M. S. (2007).
Mediation analysis. Annual Review of Psychology, 58,
593–614. doi:10.1146/annurev.psych.58.110405.085542
McGrew, K. S. (2005). The Cattell-Horn-Carroll theory of
cognitive abilities: Past, present, and future. In D. P.
Flanagan, J. L. Genshaft, & P. L. Harrison (Eds.), Contemporary intellectual assessment: Theories, tests, and issues
(pp. 136–182). New York, NY: Guilford.
Miura, I. T., Okamoto, Y., Vlahovic-Stetic, V., Kim, C. C.,
& Han, J. H. (1999). Language supports for children’s
understanding of numerical fractions: Cross-national
comparisons. Journal of Experimental Child Psychology,
74, 356–365. doi:10.1006/jecp.1999.2519
Mix, K. S., Levine, S. C., & Huttenlocher, J. (1999). Early
fraction calculation ability. Developmental Psychology, 35,
164–174. doi:10.1037/0012-1649.35.1.164
Miyake, A., Friedman, N. P., Emerson, M. J., Witzki, A.
H., & Howerter, A. (2000). The unity and diversity of
executive functions and their contributions to complex
“frontal lobe” tasks: A latent variable analysis. Cognitive
Psychology, 41, 49–100. doi:10.1006/cogp. 1999.0734.
Moses, R. P., & Cobb, C. E., Jr. (2001). Radical equations:
Civil rights from Mississippi to the algebra project. Boston,
MA: Beacon Press.
National Center for Education Statistics. (2007). The nation’s
report card. Retrieved from http://nces.ed.gov/nations
reportcard/itmrlsx/search.aspx?subject=mathematics
National Center for Education Statistics. (2009). The
nation’s report card. Retrieved from http://nces.ed.gov/
nationsreportcard/itmrlsx/search.aspx?subject=mathe
matics
National Mathematics Advisory Panel. (2008). Foundations
for success: The final report of the National Mathematics
Advisory Panel. Washington, DC: U.S. Department of
Education.
Ni, Y., & Zhou, Y. D. (2005). Teaching and learning fraction and rational numbers: The origins and implications
of whole number bias. Educational Psychologist, 40, 27–
52. doi:10.1207/s15326985ep4001_3
Pickering, S., & Gathercole, S. (2001). Working Memory
Test Battery for Children. London, UK: Psychological
Corporation.
Preacher, K. J., & Hayes, A. F. (2008). Asymptotic and
resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behavior
Research Methods, 40, 879–891. doi:10.3758/BRM.40.3.
879
Raghubar, K. P., Barnes, M. A., & Hecht, S. A. (2010).
Working memory and mathematics: A review of
developmental, individual difference, and cognitive
1476
Vukovic et al.
approaches. Learning and Individual Differences, 20, 110–
122. doi:10.1016/j.lindif.2009.10.005
Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual
and procedural knowledge of mathematics: Does one
lead to the other? Journal of Educational Psychology, 91,
175–189. doi:10.1037/0022-0663.91.1.175
Rittle-Johnson, B., Siegler, R. S., & Alibi, M. W. (2001).
Developing conceptual understanding and procedural
skill in mathematics: An iterative process. Journal of
Educational Psychology, 93, 346–362. doi:10.1037/
0022-0663.93.2.346
Sameroff, A. J., Seifer, R., Baldwin, A., & Baldwin, C.
(1993). Stability of intelligence from preschool to adolescence: The influence of social and family risk factors.
Child Development, 64, 80–97. doi:10.1111/j.1467-8624.
1993.tb02896.x
Seethaler, P. M., Fuchs, L. S., Star, J. R., & Bryant, J.
(2011). The cognitive predictors of computational skill
with whole versus rational numbers: An exploratory
study. Learning and Individual Differences, 21, 536–542.
doi:10.1016/j.lindif.2011.05.002
Siegler, R. S., & Booth, J. L. (2004). Development of
numerical estimation in young children. Child Development, 75, 428–444. doi:10.1111/j.1467-8624.2004.00684.x
Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., … Chen, M. (2012).
Early predictors of high school mathematics achievement. Psychological Science, 23, 691–697. doi:10.1177/
0956797612440101
Siegler, R. S., & Pyke, A. A. (2013). Developmental and
individual differences in understanding of fractions.
Developmental Psychology. Advance online publication.
doi:10.1037/a0031200
Slusser, E. B., Santiago, R. T., & Barth, H. C. (2013).
Developmental change in numerical estimation. Journal
of Experimental Psychology: General, 142, 193–208. doi:10.
1037/a0028560
Spence, I., & Krizel, P. (1994). Children’s perception of
proportion in graphs. Child Development, 65, 1193–1213.
doi:10.1111/j.1467-8624.1994.tb00812.x
St Clair-Thompson, H. L., & Gathercole, S. E. (2006).
Executive functions and achievements in school: Shifting, updating, inhibition, and working memory. Quarterly Journal of Experimental Psychology, 59, 745–759.
doi:10.1080/17470210500162854
Stafylidou, S., & Vosniadou, S. (2004). The development
of students’ understanding of the numerical value of
fractions. Learning and Instruction, 14, 503–518. doi:10.
1016/j.learninstruc.2004.06.015
Swanson, J., Schuck, S., Mann, M., Carlson, C., Hartman,
K., Sergeant, J., … McCleary, R. (2004). Categorical and
dimensional definitions and evaluations of symptoms of
ADHD: The SNAP and the SWAN rating scales. Retrieved
from www.adhd.net
Tatsuoka, K. K., Corter, J. E., & Tatsuoka, C. (2004). Patterns of diagnosed mathematical content and process
skills in TIMSS-R across a sample of 20 countries.
American Educational Research Journal, 41, 901–926.
doi:10.3102/00028312041004901
Vukovic, R. K., & Lesaux, N. K. (2013). The language of
mathematics: Investigating the ways language counts
for children’s mathematical development. Journal of
Experimental Child Psychology, 115, 227–244. doi:10.
1016/j.jecp.2013.02.002
Wechsler, D. (1999). Wechsler Abbreviated Scale of Intelligence. San Antonio, TX: Psychological Corporation.
Wilkinson, G. S. (1993). Wide Range Achievement Test (3rd
ed.). Wilmington: Wide Range.
Woodcock, R. W. (1997). Woodcock Diagnostic Reading
Battery. Itasca, IL: Riverside.